reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

theorem
  0 <= ||.F.||
proof
  F in BoundedFunctions X;
  then consider g be Function of X,REAL such that
A1: F=g and
A2: g|X is bounded;
A3: PreNorms g is non empty bounded_above by A2,Th17;
  consider r0 be object such that
A4: r0 in PreNorms g by XBOOLE_0:def 1;
  reconsider r0 as Real by A4;
  now
    let r be Real;
    assume r in PreNorms g;
    then ex t be Element of X st r=|.g.t.|;
    hence 0 <= r by COMPLEX1:46;
  end;
  then 0 <= r0 by A4;
  then 0 <=upper_bound PreNorms g by A3,A4,SEQ_4:def 1;
  hence thesis by A1,A2,Th20;
end;
